Nine Streets area?

The Nine Little Streets area has mostly stores and apartments, with a few restaurants. It is should be quiet at night, yet near everything you would want to see.

We've stayed a little further north (closer to the Anne Frank house) and west, but I would say that the entire Jordaan area is convenient and quiet and very beautiful. I love walking around that area.

It's one of the best places in the city center to stay.

I stayed at a hotel near De 9 Straatjes that was located right on the Herengracht. It was an excellent neighborhood. Not a loud nightlife area at all (even on weekends). Some really nice brown cafes, upscale restaurants and good boutique shopping experiences. It's also very convenient for catching various tram lines that will take you to other parts of the city, although Amsterdam itself is very walkable and fairly compact.

I'd absolutely stay in that area again on future visits. Probably the only thing I would be concerned about in your shoes would be finding a building with either a small elevator or an apartment on a lower floor as I imagine 6 people will have a pretty decent amount of luggage and the buildings in that part of town are beautiful, but old and tend to feature steep staircases. Pack light if you are staying above the 2nd floor in a walk-up building - and don't forget that 2nd Floor in Europe is really the 3rd floor in the U.S.

Ceidleh's warning about stairs applies to any place you might stay in Amsterdam. Restrictions prohibit many older buildings from having elevators, so whether you stay in a hotel or apartment, you should be prepared for stairs. And, what the Dutch call "stairs," you or I would often call "ladders" - they're very steep, with short treads and high risers. If this is an issue for you, you will have to be very careful to get a place on the ground floor or with an elevator.